Running an offline \((\gamma ,\lambda )\) -approximation algorithm on \(B_i\) , we obtain a multiset \(C_i\)  of at most \(\lambda k\)  distinct points. Let \(p^{\prime }\)  denote the element of \(C_i\)  nearest to \(\pi _i(p)\)  (this is the element of \(C_i\)  that \(p\)  gets mapped to when we pass from \(P_i \rightarrow B_i \rightarrow C_i\) ).
The following lemma implies that \(\sum _{p \in P} w(p) D(p,p^{\prime }) \le \bar{\alpha } \text{\footnotesize \textsf {OPT}}(P)\)  where \(\bar{\alpha } = \rho \alpha + 2 \rho ^2 \gamma (\alpha + 1)\) . This is an observation used widely in the literature [1]}, [2]}, but we include a proof for completeness.
